Critical exponent and blow-up rate for the omega-diffusion equations on graphs with Dirichlet boundary conditions

Abstract

In this article, we study the $\omega$-diffusion equation on a graph with Dirichlet boundary conditions $$\displaylines{ u_t(x,t)=\Delta_{\omega}u(x,t)+e^{\beta t}u^{p}(x,t), \quad (x,t)\in S\times(0,\infty), \cr u(x,t)=0, \quad (x,t)\in \partial S\times[0,\infty), \cr u(x,0)=u_0(x)\geq0, \quad x\in V, }$$ where $\Delta_{\omega}$ is the discrete weighted Laplacian operator. First, we prove the existence and uniqueness of the local solution via Banach fixed point theorem. Then, by the method of supersolutions and subsolutions we prove that the $\omega$-diffusion problem has a critical exponent $p_{\beta}$: when $p>p_{\beta}$, the solution becomes global; while when $1<p<p_{\beta}$, the solution blows up in finite time. Under appropriate hypotheses, we estimate the blow-up rate in the $L^{\infty}$-norm. Some numerical experiments illustrate our results.

Keywords

• Omega-diffusion equation
• critical exponent
• blow up
• blow-up rate
• graph